By the Perron-Frobenius theorem [1]}, if the random walk \(\mathcal {X}\)  is ergodic with a diagonalizable transition matrix \(\mathbf {P}\) , the diagonalization of \(\mathbf {P}\)  admits a simple dominant eigenvalue \(\lambda _{\max }=1\) . The other eigenvalues \(\lbrace \lambda \ne 1\rbrace \)  satisfy \(|\lambda |<1\) , which implies that all eigenvalues different of \(\lambda _{\max }\)  lie within the unit circle.
